Enhanced power spectrum analysis of short-term X-ray variability using
MAXI/GSC data
Tomofumi Kawase and Hitoshi Negoro

Department of Physics, Nihon University, 1-8 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8308, Japan
E-mail(TK): kawase@cygnus.phys.cst.nihon-u.ac.jp
Abstract
Power spectral densities (PSDs) of bright black hole candidates (BHCs) and neutron star X-ray binaries
must provide the information about, for instance, state transitions and quasi-periodic oscillations. However,a triangular window function of the eﬀective area of MAXI/GSC aﬀects the light curves of individual
sources due to the scanning observation. Suzuki (2015) established the method of evaluating resultant
PSDs correctly, and showed its usefulness for some BHCs. We modified his programs to analyze data more
eﬃciently. Scan duration to analyze in the previous method was fixed to the minimum scan duration, 40 s.
We extended the duration to use data as long as possible (≤ 200 s). As a result, we could more correctly
evaluate a PSD with a break at a lower frequency. In addition, we also improved signal-to-noise ratios ofPSDs for not-so-bright sources by cutting the edges of the window function, and evaluated its eﬀects on
the PSDs analytically.
Key words: accretion, accretion disks — X-rays: binaries — stars: individual: Cyg X-1

1. Introduction
If very bright new transients appear, MAXI/GSC data
must be important. MAXI scans the whole sky every
92 minutes. Therefore, the probability that MAXI/GSC
can discover bright new transients is higher than otherobservatories. Moreover, although observatories focusing
X-rays can not observe extremely bright sources due todetector saturation, MAXI can still do that because it
does not have mirrors to focus X-rays.
For these reasons, we must prepare the method to analyze short-term variability using MAXI/GSC data forupcoming bright new transients.
2. Power spectral density, PSD, for MAXITime duration of a scan transit for a point source is 40200 s. It depends on the direction of the source orthogonal to the scan direction. In the previous method to
obtain a power spectral density (PSD) by Suzuki (2015),
the time duration to analyze was fixed to the minimum
time duration, 40 s. We modified his programs to use
longer time duration data, which enabled us to deal withlower frequencies of the PSD. It is known that the PSD of
the black hole candidate (BHC) in the hard state shows
breaks at frequencies of 0.01-1 Hz (e.g., Pottschmidt etal. 2003). On the other hand, there is no break in thisfrequency band when the BHC is in the soft state (e.g.,Cui et al. 1997). By this improvement, it becomes easier to find the low frequency breaks, and to notice statetransitions not by energy spectra but by PSDs.
We calculate PSDs with reference to van del Klis
(1989), and normalize them according to Leahy et al.
(1983) and Miyamoto et al. (1994).
3. Eﬀects of a window function on the PSD
3.1. Time variability of the eﬀective area
The eﬀective area of GSC for a point source changes
sequentially by a collimator to restrict the direction ofincoming X-ray photons (figure 1). Therefore, a triangular window function aﬀects the light curve and its PSD.
To analyze the PSD using GSC data, we need to evaluate the eﬀect of the window function correctly.

Fig. 1. The movement of GSC in scanning observation and the time
variability of the eﬀective area for a point source. When the source
is in front of GSC, the eﬀective area reaches the maximum value.

100

100

4 10−3 0.01

0.1

1

Normalized Power

10

100.1

1

Normalized Power410−3 0.01

22

χ

χ

00−20.01−∞

Eq (2) is the Fourier transform of the window functions
in figure 2. T is the scan duration for a point source,and α is the cutting ratio for a new window function(0 < α ≤ 1). If α = 1, this equation corresponds to thewindow function of the left panel of figure 2.
{iωT2αωT2W (ω) =e4 sin222αω T4+ωT (1 − α) sin

αωT2}
(2)

4. PSD of random shot-noise data
4.1. Break at a lower frequency
We performed simulations using random shot-noise datato confirm whether it is possible to evaluate a break ofPSD at a lower frequency. A time constant of the shots
was assumed to be, for instance, 4 s. The lowest frequency of the PSD obtained by 40 s data was at 0.025
Hz. On the other hand, if we set the time duration for
100 s using our new programs, the lowest frequency was
at 0.01 Hz. We produced 48 light curves, which were atypical number of data sets for 3 days data of GSC. The
count rate of them corresponded to 400 mCrab.
Fitting the PSD by a Lorentzian model gives a breakfrequency at 0.0210 ± 0.0343 Hz. This does not tightly
constrain a break frequency because of the large error.
On the other hand, a break frequency of the PSD forthe longer time duration data was obtained at 0.0370 ±0.0029 Hz. This frequency is in good agreement withthe analytical break frequency of 0.0398 Hz for the time
constant of 4 s.

1Frequency [Hz]100.010.1

1Frequency [Hz]100.1

1Frequency [Hz]

100
100.1
0.01
10−510−410−3

Normalized Power

1

10.1
0.01
10−3

Normalized Power

Cyg_X−1_Soft_State_MJD55500−55514

1010

100

Cyg_X−1_Hard_State_MJD55228−55243

10−43.2. New window function for not-so-bright sources
In case of not-so-bright sources, the background counts
unaﬀected by the triangular window function are relatively higher than the source signal at the beginning andthe end of the scan. Therefore, we adopt a new windowfunction that cuts the edges of the triangle to remove
low signal-to-noise ratio data (figure 2). The cutting ratio should be adjusted by the brightness of the source.
Eq (1) shows the relation of a source signal x(t), awindow function w(t), and their Fourier transforms X(ω)
and W (ω). The Fourier transform of the product of x(t)and w(t) is the convolution of X(ω) and W (ω), X ∗ W .
∫ ∞x(t)w(t)eiωt dt = X(ω) ∗ W (ω)
(1)

0.1

Fig. 3. A PSD of random shot-noise aﬀected by a triangular windowfunction (Left), and a new window function (Right, α = 0.5 inEq.2). Red plots are data, and blue plots are the eﬀect of thewindow function. The PSDs are fitted by a Lorentzian.

10−5Fig. 2. Left: The previous triangular window function. Right: A newwindow function for not-so-bright sources.

0.1

1Frequency [Hz]10Fig. 4. PSDs of Cyg X-1 in the hard state (Left, MJD 55228-55243,2-20 keV), and the soft state (Right, MJD 55500-55514, 2-20keV).

4.2. PSD using a new window function
Next, we confirm that the PSD is correctly evaluated bya new window function. We prepared 100 light curvesof random shot-noise data for 100 s. The count rateof them corresponded to 400 mCrab. A time constantof the shots was 0.2 s. We also prepared light curves
removing both edges of them, and made a PSD (figure3). Since α was 0.5, the time duration of them was 50 s.
We could confirm that a large eﬀect of the windowfunction appeared in the lowest frequency bin on bothPSDs, and evaluate both of them correctly.
5. PSD of Cyg X-1
We made PSDs of the BHC, Cyg X-1, for 15 days (figure4). We used only one camera data. A PSD of the hardstate was fitted by the sum of two Lorentzians model,
and a PSD of the soft state was fitted by a power-law
model. We confirmed these results were consistent withprevious results (e.g., Pottschmidt et al. 2003).
6. Summary and Future
We modified the programs to obtain a PSD of short-term
variability using MAXI/GSC data, and confirmed thesewere useful for the actual data. We are going to analyze not-so-bright sources using this new window function and its α optimized for the background counts.
References
Cui, W. et al. 1997, The Transparent Universe, 382, 209
Leahy, D.A. et al. 1983, ApJ 272, 256-258
Miyamoto, S. et al. 1994, ApJ 435, 398-406
Pottschmidt, K. et al. 2003, A&A, 407, 1039
Suzuki, K. 2015, Master’s thesis, Nihon Universityvan del Klis, M. 1989, in Timing Neutron star, NATOASI Series C, Vol. 262, p27-70